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1、Chapter 4Application of Second Order Differential Equations in Mechanical Engineering AnalysisTai-Ran Hsu, ProfessorDepartment of Mechanical and Aerospace EngineeringSan Jose State UniversitySan Jose, California, USAME 130 Applied Engineering AnalysisChapter Outlines Review solution method of second
2、 order, homogeneous ordinary differential equations Applications in free vibration analysis- Simple mass-spring system- Damped mass-spring system Review solution method of second order, non-homogeneous ordinary differential equations - Applications in forced vibration analysis- Resonant vibration an
3、alysis- Near resonant vibration analysis Modal analysis Part 1Review Solution Method of Second Order, Homogeneous Ordinary Differential EquationsTypical form0)()()(22=+ xbudxxduadxxud(4.1)where a and b in Equation (4.1) are constantsThe solution of Equation (4.1) u(x) may be obtained by ASSUMING:u(x
4、) = emx(4.2)in which m = constant to be determinedIf the assumed solution u(x) in Equation (4.2) is valid solution, it must SATISFY theDE in Equation (4.1). That is:( ) ( )()022=+mxmxmxebdxedadxedBecause: ()mxmxemdxed222=( )mxmxmedxed=and(a)Upon substitution of the above into Equation (a) leading to
5、:( ) ( ) 02=+mxmxmxebemaemBecause emxin the expression cannot be zero (why?), we thus have:m2 + am + b = 0(4.3)Equation (4.3) is a quadratic equation, and its solution for m are:m2+ am + b = 0The quadratic equation:The TWO roots of the above quadratic equation have the forms:baamandbaam 421242122221
6、=+=(4.4)This leads to two possible solutions for the function u(x) in Equation (4.1):( )xmxmececxu2121+=(4.5)where c1and c2are the TWO arbitrary constants to be determined by TWO specified conditions, and m1and m2are expressed in Equation (4.4)Because the constant coefficients a and b in Equation (4
7、.1) are fixed with the DE, the relative magnitudes of the a, b will result in significant forms in the solution in Equation (4.5) due to the “square root” parts in the expression of m1and m2in Equation (4.4). Square root of a negative number will lead to a complex number in the solution of the DE, w
8、hich requires a special way of expressing it.We thus need to look into 3 possible cases involving relative magnitudes a and b.Case 1. a24b 0:In such case, we realize that both m1and m2 are real numbers. The solution of the Equation (4.1) is:(4.6)Case 2. a2- 4b 0:+=2/422/41222)(xbaxbaaxececexu(4.6)Ca
9、se 2: a2- 4b 0 - a Case 1 situation with 0)(6)(5)(22=+ xudxxdudxxudConsequently, we may use the standard solution in Equation (4.6) to be: +=2/422/41222)(xbaxbaaxececexuor1142= ba() ( )xxxxxececececexu32212/22/12/5 +=+=where c1and c2are arbitrary constants to be determined by given conditionsExample
10、 4.2 solve the following equation with given conditions (p. 84):0)(9)(6)(22=+ xudxxdudxxud(a)with given conditions:u(0) = 2 (b)and 0)(0=xdxxdu (c)Solution:Again by comparing Equation (a) with the typical DE in Equation (4.1), we have: a = 6 and b = 9.Further examining a24b = 62 4x9 = 36 36 = 0, lead
11、ing to special Case 3 in Equation (4.12) forthe solution:()2212221)(axaxaxexccexcecxu+=+=or ()()xxexccexccxu3212621)(+=+=(4.12)(d)Use Equation (b) for Equation (d) will yield c1= 2, leading to: () ( )xexcxu322+=Differentiating Equation (e) with condition in Equation will lead to the following manipu
12、lation:(e)()() ( ) 06232023230=+=cxcecedxxduxxxxSo, we solve for c2 = 6Hence the complete solution of Equation (a) is: ()xexxu3312)(+=Application of 2nd Order Homogeneous DEs for Mechanical Vibration AnalysisPart 2Mechanical vibration is a form of oscillatory motion of a solid or solid structure of
13、a machine.Common Sources of Mechanical Vibrations:(1) Time-varying Mechanical force or pressure.(2) Fluid induced vibration (e.g. intermittent wind, tidal waves, etc.)(3) Acoustics and ultrasonic.(4) Random movements of supports, e.g. seismic(5) Thermal, magnetic, etc.Common types of Mechanical Vibr
14、ations:0Time, tAmplitudesPeriod(1) With constant amplitudes and frequencies:0Time, tAmplitudePeriod(2) With variable amplitudes but constant frequencies0Time, tAmplitude(3) With random amplitudes and frequencies:Mechanical vibrations, in the design of mechanical systems, is normally undesirable occu
15、rrence, and engineers would attempt to either reduce it to the minimum appearance, or eliminate it completely.“Vibration Isolators” are commonly designed and used to minimize vibration of mechanicalsystems, such as:Design of vibration isolators requires analyses to quantify the amplitudes and period
16、s of the vibratory motion of the mechanical system a process called “mechanical vibration analysis”Benches for high-precision instrumentsVibration isolatorsSuspension of heavy-duty truckVibration isolatorsThe three types of mechanical vibration analyses by mechanical engineers:A. Free vibration analysis:The mechanical system (or a machine) is set to vibrate
